Why some Pyramid pairs that make 13 can never be removed

You spot a 10 and a 3 in the pyramid. They add up to 13 and both are face up, yet the game will not take them together. Tap the 10 and it says "That card is covered." The game is right: the 3 is part of what covers the 10. Here is how to spot blocked pairs at a glance, and what to do about them.

The covering rule

In Pyramid 28 face-up cards form seven rows, from one card at the top to seven at the bottom. Every card above the bottom row is overlapped by the two cards beneath it, and a card is available only when no card in the row below overlaps it. You remove two available cards whose ranks add up to 13 (A+Q, 2+J, 3+10, 4+9, 5+8 or 6+7), and a King, worth 13, is removed on its own.

So if a 6 overlaps a 7, those two can never leave together: the 7 is not available until the 6 has gone.

It reaches further than the row below

A card two rows lower does not overlap a card directly, but it overlaps one of the cards that do, so it still has to go first. Everything that must leave before a card becomes available forms a triangle beneath it: draw two lines down from the card, one through each card that overlaps it. Every card between those lines rests on it, and none of them can ever be its partner.

The triangle grows quickly as you go up:

• A card in row 6 has 2 cards resting on it.
• Row 5: 5 cards.
• Row 4: 9 cards.
• Row 3: 14 cards.
• Row 2: 20 cards.
• Row 1, the top card: all 27 others.

Cards in the same row never block each other, and cards in different rows can pair as long as neither lies in the other's triangle. The top card, though, can never pair with another pyramid card. By the time it is available everything else has gone, so it must leave with the top waste card, or on its own if it is a King: its partner has to come from the stock.

Worked example: a corner with two blocked 3s

Here is the left end of the bottom three rows:

Row 5:       10♠
Row 6:    3♥    7♣
Row 7: 6♦    K♠    3♣

These six cards hold three sums of 13: 10♠+3♥, 10♠+3♣ and 7♣+6♦. Both 3s lie inside the 10♠'s triangle, so neither can pair with it. The 6♦ also sits lower than its partner, but it is not one of the cards resting on the 7♣ (only K♠ and 3♣ are), so that pair can be made once the 7♣ is uncovered.

At the start only the bottom row is available: 6♦, K♠ and 3♣. A good order:

1. K♠ on its own. It needs no partner.
2. 3♣ with a 10 from outside the corner, for example from the waste. It must be one of the other three 10s, never the 10♠.
3. 7♣ and 6♦. The 7♣ is now uncovered, and this pair costs nothing from the stock.
4. 3♥ with another outside 10. It became available when 6♦ and K♠ left.
5. 10♠ with an outside 3. Both 3s of the corner have gone, so it needs 3♦ or 3♠.

Three outside partners clear six cards. Had you paired the 6♦ with a waste 7 at the start, just because both were available, the 7♣ would need an outside 6 as well, and the corner would cost five outside partners instead of three. A card with a partner inside the pyramid is often worth waiting for.

What to do about blocked pairs

1. Scan early. All 28 pyramid cards are face up from the deal, so you can check each card's triangle before your first move.
2. Count the partners that are left. Each rank has four cards, but a partner inside a card's triangle does not count. The 10♠ above can never use either 3 of its corner, so only 3♦ and 3♠ can take it.
3. Give stock cards to blocked cards. If a waste 10 could go with either of two 3s and one of them has a free partner elsewhere in the pyramid, use the waste card on the other.
4. Remove Kings as soon as they are available. A King only waits for the two cards that overlap it, and a King on the waste is removed on its own too.
5. Keep a partner for the top card. Near the end, make sure at least one card of its partner rank is still in the stock or waste.

Try it

These are the top three rows near the end of a deal; everything below them has gone:

         5♦
      8♠    Q♥
   K♣    A♠    8♣

Which sums of 13 can never be made here, and how many partners must come from the stock to clear all six cards?

Answer: 5♦+8♠, 5♦+8♣ and Q♥+A♠ are all blocked. Both 8s rest on the 5♦, and the A♠ overlaps the Q♥. The K♣ leaves on its own, and the other five cards each need a stock partner: a Queen for the A♠, a 5 for each 8 (only 5♣, 5♥ or 5♠ can serve), an Ace for the Q♥ and finally an 8 for the 5♦ (8♦ or 8♥).

For choosing between pairs that are all possible, read Which Pyramid pairs make 13, and look up stock, waste and other terms in the glossary.